The Cycloidal Curve

In order to simulate the rolling action of two discs, we must
consider the path traced by a point on the edge (circumference) of
one of the discs.

First consider the path traced by a point on the edge of a circle that
rotates in a stationary position. In this drawing, the
point is where the small line meets the edge of the circle. The path is
obviously that of a circle.

Now consider the path traced by this point when the circle is
rolling on a horizontal plane. (I have deliberately exaggerated the
horizontal movement of the circle in the next drawing to illustrate
more clearly the path of the point on the circle.)

If you draw a line to trace the path of the point as the circle rotates,
the result is called a Cycloidal Curve.

This curve is inaccurate in its shape because of the way I created
these drawings, so below is a much more accurate Cycloidal
Curve, generated in a chart of the horizontal and vertical
coordinates of the path of the point as the circle rotates. The chart
is used here to create the graph below (with Rotation in Degrees on the X axis and Displacement in Inches on the Y axis).